# 3-Way QuickSort (Dutch National Flag)

In simple QuickSort algorithm, we select an element as pivot, partition the array around pivot and recur for subarrays on left and right of pivot.
Consider an array which has many redundant elements. For example, {1, 4, 2, 4, 2, 4, 1, 2, 4, 1, 2, 2, 2, 2, 4, 1, 4, 4, 4}. If 4 is picked as pivot in Simple QuickSort, we fix only one 4 and recursively process remaining occurrences.

The idea of 3 way QuickSort is to process all occurrences of pivot and is based on Dutch National Flag algorithm.

In 3 Way QuickSort, an array arr[l..r] is divided in 3 parts:
a) arr[l..i] elements less than pivot.
b) arr[i+1..j-1] elements equal to pivot.
c) arr[j..r] elements greater than pivot.

Below is C++ implementation of above algorithm.

// C++ program for 3-way quick sort
#include <bits/stdc++.h>
using namespace std;

/* This function partitions a[] in three parts
a) a[l..i] contains all elements smaller than pivot
b) a[i+1..j-1] contains all occurrences of pivot
c) a[j..r] contains all elements greater than pivot */
void partition(int a[], int l, int r, int &i, int &j)
{
i = l-1, j = r;
int p = l-1, q = r;
int v = a[r];

while (true)
{
// From left, find the first element greater than
// or equal to v. This loop will definitely terminate
// as v is last element
while (a[++i] < v);

// From right, find the first element smaller than or
// equal to v
while (v < a[--j])
if (j == l)
break;

// If i and j cross, then we are done
if (i >= j) break;

// Swap, so that smaller goes on left greater goes on right
swap(a[i], a[j]);

// Move all same left occurrence of pivot to beginning of
// array and keep count using p
if (a[i] == v)
{
p++;
swap(a[p], a[i]);
}

// Move all same right occurrence of pivot to end of array
// and keep count using q
if (a[j] == v)
{
q--;
swap(a[j], a[q]);
}
}

// Move pivot element to its correct index
swap(a[i], a[r]);

// Move all left same occurrences from beginning
j = i-1;
for (int k = l; k < p; k++, j--)
swap(a[k], a[j]);

// Move all right same occurrences from end
i = i+1;
for (int k = r-1; k > q; k--, i++)
swap(a[i], a[k]);
}

// 3-way partition based quick sort
void quicksort(int a[], int l, int r)
{
if (r <= l) return;

int i, j;

// Note that i and j are passed as reference
partition(a, l, r, i, j);

// Recur
quicksort(a, l, j);
quicksort(a, i, r);
}

// A utility function to print an array
void printarr(int a[], int n)
{
for (int i = 0; i < n; ++i)
printf("%d  ", a[i]);
printf("\n");
}

// Driver program
int main()
{
int a[] = {4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4};
int size = sizeof(a) / sizeof(int);
printarr(a, size);
quicksort(a, 0, size - 1);
printarr(a, size);
return 0;
}

Output:

4  9  4  4  1  9  4  4  9  4  4  1  4
1  1  4  4  4  4  4  4  4  4  9  9  9

Thanks to Utkarsh for suggesting above implementation.

Another Implementation using Dutch National Flag Algorithm

// C++ program for 3-way quick sort
#include <bits/stdc++.h>
using namespace std;

void swap(int *a, int *b)
{
int temp = *a;
*a = *b;
*b = temp;
}

// A utility function to print an array
void printarr(int a[], int n)
{
for (int i = 0; i < n; ++i)
printf("%d ", a[i]);
printf("\n");
}

/* This function partitions a[] in three parts
a) a[l..i] contains all elements smaller than pivot
b) a[i+1..j-1] contains all occurrences of pivot
c) a[j..r] contains all elements greater than pivot */

//It uses Dutch National Flag Algorithm
void partition(int a[], int low, int high, int &i, int &j)
{
// To handle 2 elements
if (high - low <= 1)
{
if (a[high] < a[low])
swap(&a[high], &a[low]);
i = low;
j = high;
return;
}

int mid = low;
int pivot = a[high];
while (mid <= high)
{
if (a[mid]<pivot)
swap(&a[low++], &a[mid++]);
else if (a[mid]==pivot)
mid++;
else if (a[mid]>pivot)
swap(&a[mid], &a[high--]);
}

//update i and j
i = low-1;
j = mid; //or high-1
}

// 3-way partition based quick sort
void quicksort(int a[], int low, int high)
{
if (low>=high) //1 or 0 elements
return;

int i, j;

// Note that i and j are passed as reference
partition(a, low, high, i, j);

// Recur two halves
quicksort(a, low, i);
quicksort(a, j, high);
}

// Driver program
int main()
{
int a[] = {4, 9, 4, 4, 1, 9, 4, 4, 9, 4, 4, 1, 4};
//int a[] = {4, 39, 54, 14, 31, 89, 44, 34, 59, 64, 64, 11, 41};
//int a[] = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
//int a[] = {91, 82, 73, 64, 55, 46, 37, 28, 19, 10};
//int a[] = {4, 9, 4, 4, 9, 1, 1, 1};
int size = sizeof(a) / sizeof(int);

printarr(a, size);
quicksort(a, 0, size - 1);
printarr(a, size);
return 0;
}

Output:

4  9  4  4  1  9  4  4  9  4  4  1  4
1  1  4  4  4  4  4  4  4  4  9  9  9

Thanks Aditya Goel for this implementation.

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